(a) Technical Field
The present disclosure relates generally to projectiles for use in firearms, and more particularly, to projectiles for use in firearms having enhanced ballistic efficiency.
(b) Background Art
Early in the 20th century, as the first aircrafts were being built and launched, engineers and mathematicians worked to optimize the flight of such aircrafts, as well as propulsion methods, controls, and the like. One of the shapes that was created through mathematical analysis is known as the Sears-Haack shape. The Sears-Haack shape—derived from the work of William Sears and Wolfgang Haack—is regarded as exhibiting the minimum theoretical wave drag on a given body at high supersonic speeds. A Sears-Haack body is axisymmetric, decreasing smoothly in opposite directions from a maximum diameter at its center to a sharply pointed tip at each end, resembling somewhat the shape of a football. Sears-Haack bodies are reliant upon the Prandtl-Glauert transformation to solve the mathematical singularity that occurs from compression shock (and subsequent wave drag) generated at near-Mach and supersonic speeds.
Any given Sears-Haack body can be mathematically adjusted according to the preference of its designer. Specifically, the dimensions of Sears-Haack shapes, known also as the Haack Series shapes, can be fine-tuned by selecting the parameters in Equations 1 and 2:
                    y        =                              R                          π                                ⁢                                    θ              -                                                sin                  ⁡                                      (                                          2                      ⁢                      θ                                        )                                                  2                            +                              C                ⁢                                                                  ⁢                                  sin                  3                                ⁢                θ                                                                        [                  Equation          ⁢                                          ⁢          1                ]                                          θ          =                      arccos            ⁡                          (                              1                -                                                      2                    ⁢                                                                                  ⁢                    x                                    L                                            )                                      ,                            [                  Equation          ⁢                                          ⁢          2                ]            where L represents a length of a body, R represents a maximum radius of the body, and C affects the shape of the body.
It can be seen that the Sears-Haack equations allow for a continuous set of shapes determined by the values of L, R, and C. A non-dimensional variable created from L/R is known as the “fineness ratio,” or sometimes “aspect ratio,” which allows designers to carry a shape of the nose across different bodies (e.g., to investigate scaling effects resulting from changing parameters during design of a body). However, the value of C is the driving force in determining body shape and the defining feature to transform a Sears-Haack shape into any of the specialty Haack shapes.
Two values of C have particular significance for optimizing the aerodynamic design of a body: C=0, signifying minimum drag for a given length and diameter, also known as an “LD-Haack,” and C=⅓, signifying minimum drag for a given length and volume, also known as an “LV-Haack.” The LD-Haack, in particular, or commonly referred to as the “von Kármán”—named after Theodore von Kármán who developed an adaptation of the Sears-Haack to minimize wave drag on objects travelling at supersonic speeds—has been adopted to optimize the aerodynamic performance of various objects meant to travel through a compressible fluid medium. Not surprisingly, the von Kármán shape is heavily used in current-day aerospace flight vehicles due to its capacity for minimizing drag occurring at the nose cone section of aircrafts. The applicability of von Kármán is not limited to aircrafts, however, as it is possible to implement the von Kármán shape in other types of travelling objects, including firearm projectiles.